基于改进牛顿法的梁结构拓扑优化方法  

Topology Optimization Method of Beam Structure Based on Improved Newton-Raphson Algorithm

作　　者：商鹏 张海瑞[1] 阳加远 王辉雄 洪东跑[1] SHANG Peng;ZHANG Hai-rui;YANG Jia-yuan;WANG Hui-xiong;HONG Dong-pao(China Academy of Launch Vehicle Technology,Beijing 100076,China)

机构地区：[1]中国运载火箭技术研究院,北京100076

出　　处：《强度与环境》2023年第2期40-46,共7页Structure & Environment Engineering

基　　金：国家自然科学基金(U20B2028)资助。

摘　　要：面向飞行器结构轻量化、高刚度的设计需求,为改进传统结构拓扑优化方法存在初始值选择条件严苛、收敛速度慢等问题,将牛顿-拉普森算法和外点惩罚函数法相结合,研究提出一种基于改进牛顿法的梁模型结构拓扑优化方法。结合飞行器翼舵、支架和端框等支撑结构的特点,建立基于变密度法的结构拓扑优化模型,并采用牛顿-拉普森算法进行模型参数求解。引入外点惩罚函数法对牛顿-拉普森算法进行改进,通过综合考虑优化过程中梯度方向与初始点选取的影响,构建惩罚函数代替拉格朗日函数。进一步,通过考虑单元节点位移与相对密度关系,求解更为精确的海森矩阵,有效提升算法的收敛效率和稳定性。以飞行器结构中典型梁结构作为研究对象,开展案例应用研究。结果表明该方法能够得到边界清晰、结果稳定性高的结构拓扑优化模型,且优化精度与效率较传统方法高,可为飞行器结构轻量化设计提供支持。In order to improve the design requirements of lightweight and high stiffness of aircraft structure,a structural topology optimization method based on the improved Newton's method is proposed by combining the Newton-Raphson algorithm and the outer point penalty function method to improve the problems of strict initial value selection and slow convergence of traditional structural topology optimization methods.The technique is oriented to the design requirements of light weight and high stiffness,establishes a topology optimization model based on the variable density method,and solves the penalty function by considering the influence of gradient direction and initial point selection in the optimization process.For this model,the exact Hessian matrix is further solved to solve the problem of numerical instability and harsh initial value selection by considering the relationship between unit node displacement and relative density.The typical plate and beam structures commonly found in aircraft structures are used as design objects to study the case application of the above method.The example results show that the method can obtain an optimization model with clear boundaries and high stability of results,which improves the optimization accuracy and efficiency and provides support for lightweight design of aircraft.

关 键 词：拓扑优化 外点惩罚函数法 牛顿-拉普森算法 海森矩阵 

分 类 号：V414.1[航空宇航科学与技术—航空宇航推进理论与工程]